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Mach configuration in pseudo-stationary compressible flow

Author: Shuxing Chen
Journal: J. Amer. Math. Soc. 21 (2008), 63-100
MSC (2000): Primary 35L65, 35L67, 76N10
Published electronically: March 5, 2007
MathSciNet review: 2350051
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Abstract: This paper is devoted to studying the local structure of Mach reflection, which occurs in the problem of the shock front hitting a ramp. The compressible flow is described by the full unsteady Euler system of gas dynamics. Because of the special geometry, the motion of the fluid can be described by self-similar coordinates, so that the unsteady flow becomes a pseudo-stationary flow in this coordinate system. When the slope of the ramp is less than a critical value, the Mach reflection occurs. The wave configuration in Mach reflection is composed of three shock fronts and a slip line bearing contact discontinuity. The local existence of a flow field with such a configuration under some assumptions is proved in this paper. Our result confirms the reasonableness of the corresponding physical observations and numerical computations in Mach reflection. In order to prove the result, we formulate the problem to a free boundary value problem of a pseudo-stationary Euler system. In this problem two unknown shock fronts are the free boundary, and the slip line is also an unknown curve inside the flow field. The proof contains some crucial ingredients. The slip line will be transformed to a fixed straight line by a generalized Lagrange transformation. The whole free boundary value problem will be decomposed to a fixed boundary value problem of the Euler system and a problem to updating the location of the shock front. The Euler system in the subsonic region is an elliptic-hyperbolic composite system, which will be decoupled to the elliptic part and the hyperbolic part at the level of principal parts. Then some sophisticated estimates and a suitable iterative scheme are established. The proof leads to the existence and stability of the local structure of Mach reflection.

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Additional Information

Shuxing Chen
Affiliation: School of Mathematical Sciences, Fudan University, and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University, Ministry of Education), Shanghai 200433, People’s Republic of China

Keywords: Euler system, Mach configuration, pseudo-stationary flow, free boundary problem, elliptic-hyperbolic composed system
Received by editor(s): November 14, 2005
Published electronically: March 5, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.